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Given a totally positive matrix, can one insert a line (row or column) between two given lines while maintaining total positivity? This question was first posed and solved by Johnson and Smith who gave an algorithm that results in one possible line insertion. In this work we revisit this problem. First we show that every totally positive matrix can be associated to a certain vertex-weighted graph in such a way that the entries of the matrix are equal to sums over certain paths in this graph. We call this graph a scaffolding of the matrix. We then use this to give a complete characterization of all possible line insertions as the strongly positive solutions to a given homogeneous system of linear equations.
A matrix $A$ is totally positive (or non-negative) of order $k$, denoted $TP_k$ (or $TN_k$), if all minors of size $leq k$ are positive (or non-negative). It is well-known that such matrices are characterized by the variation diminishing property tog
Given a nonnegative matrix $A$, can you find diagonal matrices $D_1,~D_2$ such that $D_1AD_2$ is doubly stochastic? The answer to this question is known as Sinkhorns theorem. It has been proved with a wide variety of methods, each presenting a variet
For any primitive matrix $Minmathbb{R}^{ntimes n}$ with positive diagonal entries, we prove the existence and uniqueness of a positive vector $mathbf{x}=(x_1,dots,x_n)^t$ such that $Mmathbf{x}=(frac{1}{x_1},dots,frac{1}{x_n})^t$. The contribution of
The composition operators preserving total non-negativity and total positivity for various classes of kernels are classified, following three themes. Letting a function act by post composition on kernels with arbitrary domains, it is shown that such
Not every positive functional defined on bi-variate polynomials of a prescribed degree bound is represented by the integration against a positive measure. We isolate a couple of conditions filling this gap, either by restricting the class of polynomi